Coherent states for quadratic Hamiltonians
Alonso Contreras-Astorga, David J Fernandez C, Mercedes Velazquez
aa r X i v : . [ qu a n t - ph ] D ec Coherent states for quadratic Hamiltonians
Alonso Contreras-Astorga, David J. Fern´andez C. and Mercedes Vel´azquezDepartamento de F´ısica, Cinvestav, A.P. 14-740,07000 M´exico D.F., Mexico
Abstract
The coherent states for a set of quadratic Hamiltonians in the trap regime are con-structed. A matrix technique which allows to identify directly the creation and annihilationoperators will be presented. Then, the coherent states as simultaneous eigenstates of theannihilation operators will be derived, and they are going to be compared with those at-tained through the displacement operator method. The corresponding wave function willbe found, and a general procedure for obtaining several mean values involving the canon-ical operators in these states will be described. The results will be illustrated through theasymmetric Penning trap.
PACS: 03.65.Ge, 03.65.Sq, 37.10.Ty, 37.30.+i
As it was shown by Schr¨odinger in 1926 for the harmonic oscillator, the quasi-classical statesare important for the description of physical systems in the classical limit (see e.g. [1]). Thecatchy term coherent states (CS) was used for the first time by Glauber long after, when studyingelectromagnetic correlation functions [2,3]. With this application it was realized that the CS areuseful as well in the intrinsically quantum domain. Indeed, the CS approach is nowadays widelyemployed for dealing with quantum physical systems. According to Glauber there are threeequivalent ways to construct the CS for the harmonic oscillator. The first one is to define themas eigenstates of the annihilation operator. The second one is to build the CS through the actionof a displacement operator onto the ground state. The third way is to consider them as quantumstates having a minimum Heisenberg uncertainty relationship. These three properties can beused as definitions to build the CS for systems different from the harmonic oscillator. However,it is noteworthy that each of them leads to sets of CS which do not coincide in general [4–13].In fact, even for the harmonic oscillator the third definition does not produce just the standardCS, since it includes as well the so-called squeezed states [14, 15].In spite of its long term life, from time to time there are some advances which maintainthe subject alive. This is the case, e.g., of the recently discovered coherent states for a chargedparticle inside an ideal Penning trap [16]. Since the corresponding Hamiltonian is quadratic inthe position and momentum operators, one would expect that the CS appear as a generalizeddisplacement operator acting onto the corresponding ground state. However, it is worth to noticethat the Penning trap Hamiltonian does not have any ground state at all, since it is not a positively1efined operator. Despite, it was possible to implement in a simple way the corresponding CSconstruction. Thus, we need to take into account this Hamiltonian property when studying thecoherent states of general systems.In this article we are going to address the CS construction for systems characterized bya certain set of quadratic Hamiltonians. The CS will be built up as simultaneous eigenstatesof the corresponding annihilation operators, and also by applying a generalized displacementoperator onto an appropriate extremal state. We will see that in the case of a positively definedHamiltonian this extremal state will coincide with the ground state. In order to perform the CSconstruction, we need to find first the annihilation and creation operators. This task will be doneby using a matrix technique, which generalizes the one employed in [16] (see also [17–20]). Inthis way, we will simply and systematically identify the characteristic algebra of the involvedHamiltonians. Our procedure represents a generalization to dimensions greater than one of thestandard technique to deal with the harmonic oscillator, which is closely related to the wellknown factorization method (see, e.g., [21, 22]).The paper is organized as follows. In section 2 we will introduce a detailed recipe forsystematically obtaining the annihilation and creation operators for quadratic Hamiltonians inthe trap regime. The coherent states derivation shall be elaborated in section 3, while in section4 we will address the completeness of this set of CS, we shall obtain the mean values of severalimportant physical quantities and the time evolution of these states. We are going to apply ourgeneral results to an asymmetric Penning trap in section 5, and our conclusions will be presentedat section 6.
Along this work we are going to consider a general set of n -dimensional quadratic Hamiltoniansof the form H = 12 η T B η, (1)where B is a n × n real constant symmetric matrix, η = (cid:0) ~X, ~P (cid:1) T , and ~X, ~P are the n -dimensional coordinate and momentum operators in the Schr¨odinger picture satisfying thecanonical commutation relationships [ X i , P j ] = iδ ij (notice that a system of units such that ~ = 1 will be used throughout this paper). The time evolution of the operator vector η ( t ) = U † ( t ) ηU ( t ) in the Heisenberg picture is governed by dη ( t ) dt = U † ( t )[ iH, η ] U ( t ) = U † ( t ) Λ ηU ( t ) = Λ η ( t ) , (2) U ( t ) being the evolution operator of the system such that U (0) = , and Λ = JB , where J isthe well known n × n matrix J = (cid:18) n − n (cid:19) , (3)satisfying J T = − J , J = − n , det( J ) = 1 , (4)in which m represents the m × m identity matrix. The solution of Eq.(2) is given by η ( t ) = e Λ t η (0) = e Λ t η. (5)2n order to identify the annihilation and creation operators of H , we need to find the rightand left eigenvectors of Λ . Since in general Λ is non-hermitian, its right and left eigenvectorsare not necessarily adjoint to each other.Let us consider in the first place the n -th order characteristic polynomial of Λ , P ( λ ) =det( Λ − λ ) = det( JB − λ ) . Using Eqs.(4) and the fact that B is a symmetric matrix, we obtain P ( λ ) = det( JB − λ ) = det[( JB − λ ) T ] = det( JB + λ ) = P ( − λ ) . (6)This means that if λ is an eigenvalue of Λ , then − λ also will be. Throughout this work we aregoing to denote the eigenvalues of Λ as λ k and − λ k , taking λ k in the way Re( λ k ) > or Im( λ k ) > if Re( λ k ) = 0 , k = 1 , . . . , n. (7)Let us label as u ± k and f ± k the right and left eigenvectors associated to the eigenvalues ± λ k respectively, i.e., Λ u ± k = ± λ k u ± k , f ± k Λ = ± λ k f ± k . (8)Notice that both u ± k and f ± k can be determined from Eq.(8) up to arbitrary factors. Part of thisarbitrariness will be eliminated by imposing two requirements. The first one is that the rightand left eigenvectors be dual to each other, namely, f rj u r ′ k = δ jk δ rr ′ , (9)with j, k = 1 , . . . , n and r, r ′ = + , − . The second condition, which is needed in order to recoverthe standard annihilation and creation operators for the one-dimensional harmonic oscillator, isto ask that the left eigenvectors f ± k involved in the commutators [ f − k η, f + k η ] = γ k , γ k ∈ C , (10)are such that | γ k | = 1 . (11)In this paper we are going to discuss just the case in which there are no degeneracies in theeigenvalues ± λ k so that the identity matrix n can be expanded as (see [23]) n = n X k =1 ( u + k ⊗ f + k + u − k ⊗ f − k ) , (12)with ⊗ representing tensor product. Then we get η ( t ) = e Λ t " n X k =1 ( u + k ⊗ f + k + u − k ⊗ f − k ) η = n X k =1 [ e λ k t u + k ⊗ f + k η + e − λ k t u − k ⊗ f − k η ]= n X k =1 ( e λ k t u + k L + k + e − λ k t u − k L − k ) , (13)where L ± k ≡ f ± k η . 3t is worth to point out that, in the classical case, L ± k represent c -numbers which are relatedwith the initial conditions. The time dependence of η ( t ) is determined essentially by the λ k -values, which are complex in general. If Re( λ k ) = 0 it can be seen that one of the two involvedexponentials of the k -th term in the previous relation diverges as t increases and, thus, theclassical motion will be in general unbounded (see [24–27]). The only way in which this doesnot happen is that all the eigenvalues be purely imaginary so that the corresponding exponentialswill just induce oscillations in time, and therefore in this case the classical evolution of thevector η ( t ) will remain always bounded.On the other hand, in the quantum regime the L ± k are linear operators in the canonicalvariables ~X, ~P . It is straightforward to show that their commutators with H reduce to [ H, L ± k ] = ∓ iλ k L ± k . (14)In addition, it turns out that [ L − j , L − k ] = [ L + j , L + k ] = 0 , [ L − j , L + k ] = 0 , k = j. (15)However, [ L − k , L + k ] = γ k = 0 , k = 1 , . . . , n. (16)Eq.(14) implies that L ± k behave, at least formally, as ladders operators for the eigenvectors of H ,changing its eigenvalues by ∓ iλ k . However, this statement has to be managed carefully sinceit could happen that the action of L ± k onto an eigenvector of H produces something which doesnot belong to the domain of H , and in this case we would not get new eigenvectors of H . In thenext section we will explore an interesting situation (the trap regime of our systems) for whichthe application of L ± k onto an eigenvector of H produce a new one with different eigenvalue.Let us point out that Eqs.(14-16) imply that H can be expressed in a simple way in terms of { L ± k , k = 1 , . . . , n } [24]. This is a consequence of the following general theorem: Theorem . If L is an irreducible algebra of operators generated by L ± i which obey [ L + i , L + j ] =[ L − i , L − j ] = 0 , [ L − i , L + j ] = γ i δ ij with | γ i | = 1 , i, j = 1 , . . . , n , then an operator H ∈ L whichfulfills relation (14) can be written as H = n X k =1 (cid:16) − iλ k γ k (cid:17) L + k L − k + g , (17)where g ∈ C . Demonstration . Actually, due to Eqs.(14-16) it turns out that g ≡ H − n X k =1 (cid:16) − iλ k γ k (cid:17) L + k L − k commutes with L ± k for all k , thus with any function of them, and so g must be a c -number. (cid:3) From now on we are going to discard situations such that
Re( λ k ) = 0 for some k = 1 , . . . , n ,restricting ourselves to cases in which the λ k are purely imaginary for all k , i.e., we stick to thetrap regime of our systems. 4 .1 Algebraic structure of H in the trap regime Let us suppose that λ k = iω k with ω k > , k = 1 , . . . , n . Hence, λ ∗ k = − λ k , and since Λ isreal, without loosing generality we can choose f − k = ( f + k ) ∗ , u − k = ( u + k ) ∗ . (18)Since L ± k are linear combinations of the hermitian components of η ( X , . . . , X n , P , . . . , P n ),it turns out that ( L ± k ) † = L ∓ k . (19)Note moreover that γ ∗ k = γ k , i.e., γ k ∈ R , and the use of Eq.(11) implies that γ k = ± .Summarizing these results, Eq.(14) becomes in this case [ H, L ± k ] = ± ω k L ± k , (20)i.e., the modifications suffered by the eigenvalues of H through the action of the ladder operators L ± k are given by the real quantities ± ω k . In addition, H is factorized as (see Eq.(17)) H = n X k =1 γ k ω k L + k L − k + g , g ∈ R . (21)Notice that the previous summation either involves terms for which γ k = 1 , which are of theoscillator kind since they are positively defined, or terms with γ k = − which are of the anti-oscillator type since they are negatively defined. Thus, it is natural to define a global algebraicstructure for our system (see, e.g., [16, 28]), which is independent of the spectral details buthas to do with the fact that any quadratic Hamiltonian in the trap regime can be expressed interms of several independent oscillators, some of them indeed being anti-oscillators (compareEq.(21)). This global structure is characterized mathematically by identifying n sets of number,annihilation and creation operators of the system, { N k , B k , B † k } , k = 1 , . . . , n , in the way: B k = L − k , B † k = L + k , for γ k = 1 ,B k = L + k , B † k = L − k , for γ k = − , (22) N k = B † k B k , k = 1 , . . . , n, so that the standard commutation relationships are satisfied, [ B j , B † k ] = δ jk , [ B j , B k ] = [ B † j , B † k ] = 0 , (23) [ N k , B k ] = − B k , [ N k , B † k ] = B † k , j, k = 1 , . . . , n. Let us construct now a basis {| n , . . . , n n i , n j = 0 , , , . . . , j = 1 , , . . . , n } of commoneigenstates of { N , . . . , N n } (the Fock states) N j | n , . . . , n n i = n j | n , . . . , n n i , j = 1 , . . . , n, (24)starting from an extremal state | , . . . , i , which is annihilated simultaneously by B , . . . , B n : B j | , . . . , i = 0 , j = 1 , . . . , n. (25)5f we assume that | , . . . , i is normalized, it turns out that: | n , . . . , n n i = B † n . . . B † nn n | , . . . , i√ n ! · · · n n ! . (26)Moreover, B j and B † j , j = 1 , . . . , n , act onto | n , . . . , n n i in a standard way: B j | n , . . . , n j − , n j , n j +1 , . . . , n n i = √ n j | n , . . . , n j − , n j − , n j +1 , . . . , n n i ,B † j | n , . . . , n j − , n j , n j +1 , . . . , n n i = p n j +1 | n , . . . , n j − , n j +1 , n j +1 , . . . , n n i . (27)Now, in terms of the operators { B j , B † j , j = 1 , . . . , n } our Hamiltonian is expressed by H = n X k =1 γ k ω k B † k B k + g ′ . (28)It is clear that the Fock states | n , . . . , n n i are eigenstates of H with eigenvalues E n ,...,n n = γ ω n + . . . + γ n ω n n n + g ′ ≡ E ( n , . . . , n n ) . In particular, the extremal state | , . . . , i has eigenvalue E ,..., = g ′ . In case that γ k = 1 for all k , then H − g ′ will be a positivelydefined operator, and the extremal state | , . . . , i will become the ground state for our system,associated to the lowest eigenvalue E ,..., = g ′ of H . On the other hand, if there is at least oneindex j for which γ j = − , then H − g ′ will not be positively defined, since the corresponding j -th term is of inverted oscillator type, and the state | , . . . , i will not be a ground state for oursystem (however it keeps its extremal nature since it is always annihilated by the n operators B j , j = 1 , . . . , n ).Following [16, 28] it is straightforward to see that, besides the global algebraic structure,there is an intrinsic algebraic structure for our system, characterized by the existing relationshipbetween the Hamiltonian H and the n number operators N k : H = E ( N , . . . , N n ) = n X k =1 γ k ω k N k + g ′ . (29)As in the examples discussed in [16, 28], it turns out that this intrinsic algebraic structure isresponsible for the specific spectrum of our Hamiltonian. On the other hand, the global alge-braic structure arises from the existence of the n independent oscillator modes for H , each onecharacterized by the standard generators { N j , B j , B † j } , j = 1 , . . . , n . This global behavior al-lows us to identify in a natural way the extremal state | , . . . , i ≡ | i , which plays the role ofa ground state although it does not necessarily has a minimum energy eigenvalue. Moreover,the very existence of the extremal state | i is guaranteed by a theorem [24] ensuring that if theoperators { B , . . . , B n } obey the commutation relations given by Eq.(23), then the system ofpartial differential equations h ~x | B j | , . . . , i = h ~x | B j | i = 0 , j = 1 , . . . , n, (30)has the square integrable solution φ ( ~x ) = h ~x | i = ce − a ij x i x j = ce − ( ~x T a ~x ) , (31)6ith c being a normalization factor. In this wave function, a = ( a ij ) represents a symmetricmatrix whose complex entries are found by solving the system of equations (30), leading to a ~α j = ~β j , j = 1 , . . . , n, (32)where ~α j and ~β j are obtained by expressing B j and B † j as B j = i ~P · ~α j + ~X · ~β j , B † j = − i~α † j · ~P + ~β † j · ~X, j = 1 , . . . , n. (33)The wave functions for the other Fock states can be found from Eq.(26). Once our Hamiltonian has been expressed appropriately in terms of annihilation and creationoperators, we can develop a similar treatment as for the harmonic oscillator to build up thecorresponding coherent states. Here we are going to construct them either as simultaneouseigenstates of the annihilation operators of the system or as the ones resulting from acting theglobal displacement operator onto the extremal state.
In the first place let us look for the annihilation operator coherent states (AOCS) as commoneigenstates of the B j ’s: B j | z , . . . , z n i = z j | z , . . . , z n i , z j ∈ C , j = 1 , . . . , n. (34)Following a standard procedure, let us expand them in the basis {| n , . . . , n n i} : | z , . . . , z n i = ∞ X n ,...,n n =0 c n ,...,n n | n , . . . , n n i . (35)By imposing now that Eq.(34) is satisfied, the following recurrence relationships are obtained, c n ,...,n j ,...,n n = z j √ n j c n ,...,n j − ,...,n n , j = 1 , . . . , n, (36)which, when iterated, lead to c n ,...,n j ,...,n n = z n j j p n j ! c n ,..., ,...,n n , j = 1 , . . . , n. (37)Hence, it turns out that c n ,...,n n = z n . . . z n n n √ n ! · · · n n ! c ,..., , (38)where c ,..., is to be found from the normalization condition. Thus the normalized AOCSbecome finally: | z , . . . , z n i = exp − n X j =1 | z j | ! ∞ X n ,...,n n =0 z n . . . z n n n | n , . . . , n n i√ n ! · · · n n ! , (39)up to a global phase factor. 7 .2 Displacement Operator Coherent States (DOCS) The displacement operator for the j -th oscillator mode of the Hamiltonian reads D j ( z j ) = exp (cid:16) z j B † j − z ∗ j B j (cid:17) . (40)By using the BCH formula it turns out that D j ( z j ) = exp (cid:18) − | z j | (cid:19) exp (cid:16) z j B † j (cid:17) exp (cid:0) − z ∗ j B j (cid:1) . (41)Now, the global displacement operator is given by: D ( z ) ≡ D ( z , . . . , z n ) = D ( z ) · · · D n ( z n ) , (42)where z denotes the complex variables z , . . . , z n associated to the n oscillator modes.Let us obtain now the displacement operator coherent states (DOCS) | z i from applying D ( z ) onto the extremal state | , . . . , i ≡ | i : | z i = D ( z ) | i = exp − n X j =1 | z j | ! ∞ X n ,...,n n =0 z n . . . z n n n | n , . . . , n n i√ n ! · · · n n ! . (43)Notice that the AOCS and the DOCS are the same (compare Eqs.(39) and (43)). In order to find the wave functions of the coherent states previously derived, we employ that [ z j B † j − z ∗ j B j , z k B † k − z ∗ k B k ] = 0 ∀ j, k . Thus: D ( z ) = exp( z B † − z ∗ B ) · · · exp( z n B † n − z ∗ n B n )= exp[( z B † + · · · + z n B † n ) − ( z ∗ B + · · · + z ∗ n B n )] . (44)Using now Eq.(33) we can write D ( z ) = e − i ( ~ Γ · ~P − ~ Σ · ~X ) = e − i ~ Γ · ~ Σ e i~ Σ · ~X e − i~ Γ · ~P = e i ~ Γ · ~ Σ e − i~ Γ · ~P e i~ Σ · ~X , (45)where we have employed once again the BCH formula and we have taken ~ Γ = 2Re[ z ∗ ~α + . . . + z ∗ n ~α n ] , ~ Σ = − z ∗ ~β + . . . + z ∗ n ~β n ] . (46)Now, it is straightforward to find the wave function for the coherent state | z i , φ z ( ~x ) = h ~x | z i = h ~x | D ( z ) | i = e − i ~ Γ · ~ Σ e i~ Σ · ~x h ~x | e − i ~P · ~ Γ | i . (47)Since the operator ~P is the coordinate displacement generator [1], it turns out that h ~x | e − i ~P · ~ Γ = h ~x − ~ Γ | , (48)8o that φ z ( ~x ) = e − i ~ Γ · ~ Σ e i~ Σ · ~x h ~x − ~ Γ | i = e − i ~ Γ · ~ Σ e i~ Σ · ~x φ ( ~x − ~ Γ) . (49)A further calculation, using Eq.(31), leads finally to φ z ( ~x ) = e − ( ~ Γ T a + i~ Σ) · ~ Γ e ( ~ Γ T a + i~ Σ) · ~x φ ( ~x ) . (50)Once again, it becomes evident that the extremal state is important in our treatment, since itswave function determines the corresponding wave function for any other CS. Moreover, as it canbe seen from Eq.(49), the position probability density for the CS | z i becomes just a displacedversion of the corresponding one for the extremal state | i . Let us derive next the completeness relationship for the previously derived coherent states. No-tice that, from the point of view of the analysis of states in the Hilbert space of the system, thisis the most important property which our CS would have [9, 11, 29, 30]. This is the reason whyseveral authors use it as the fourth coherent state definition, considering it as the fundamentalone which will survive in time (see e.g. [11]). We are going to calculate as well some importantphysical quantities in these states.
A straightforward calculation leads to: (cid:18) π (cid:19) n Z · · · Z | z ih z | d z . . . d z n = ∞ X m ,n ,...,m n ,n n =0 | m , . . . , m n ih n , . . . , n n |√ m ! n ! · · · m n ! n n ! n Y j =1 (cid:18) π Z z m j j z ∗ j n j e −| z j | d z j (cid:19) = , (51)with being the identity operator. Thus, the coherent states {| z i} form a complete set in thestate space of the system (indeed they constitute an overcomplete set [31,32]). This implies thatany state can be expressed in terms of our coherent states, in particular, an arbitrary coherentstate, | z ′ i = (cid:18) π (cid:19) n Z · · · Z | z ih z | z ′ i d z . . . d z n , (52)where the reproducing kernel h z | z ′ i is given by h z | z ′ i = exp " − n X j =1 (cid:0) | z j | − z ∗ j z ′ j + | z ′ j | (cid:1) . (53)This means that, in general, our coherent states are not orthogonal to each other. Indeed, noticethat inside our infinite set of coherent states only the extremal state of the system, | i ≡ | z = i = | , . . . , i , is also an eigenstate of the Hamiltonian.9 .2 Mean values of some physical quantities in a CS Now we can calculate easily the mean values h X j i z ≡ h z | X j | z i , h P j i z ≡ h z | P j | z i , j = 1 , . . . , n ,in a given coherent state | z i = | z , . . . , z n i , as well as its mean square deviation in terms of thecorresponding results for the extremal state | i . To do that, let us analyze first how the operators X j , X j , P j , P j are transformed under D ( z ) . By using Eqs.(45,46) it is straightforward to showthat: D † ( z ) X nj D ( z ) = ( X j + Γ j ) n , D † ( z ) P nj D ( z ) = ( P j + Σ j ) n , n = 1 , , . . . (54)where we have used that, for an operator A which commutes with [ A, B ] , it turns out that e A Be − A = B + [ A, B ] ⇒ e A B n e − A = ( B + [ A, B ]) n , n = 1 , , . . . . (55)Thus, a straightforward calculation leads to h X j i z ≡ h z | X j | z i = h | D † ( z ) X j D ( z ) | i = h X j i + Γ j . (56)On the other hand, h X j i z = h X j i + 2Γ j h X j i + Γ j . (57)Hence, (∆ X j ) z = h X j i z − h X j i z = h X j i − h X j i = (∆ X j ) . (58)Working in a similar way for P j , it is obtained h P j i z = h P j i + Σ j , h P j i z = h P j i + 2Σ j h P j i + Σ j . (59)Then we have as well that (∆ P j ) z = (∆ P j ) , (60)i.e., the mean square deviations of X j and P j in the CS | z i are independent of z .In order to end up this calculation, the mean values h X j i , h P j i , h X j i , and h P j i for j = 1 , . . . , n are required. Let us describe now the procedure to find these n quantities. Thefirst n , h X j i , h P j i , can be easily found by recalling the definitions of B k , B † k (see section 2.1)and using the fact that their mean values in the extremal state | i always vanish for k = 1 , . . . , n : h B k i = h B † k i = 0 , k = 1 , . . . , n. (61)This is equivalent to the following linear system of n homogeneous equations f − k h η i = f + k h η i = 0 , k = 1 , . . . , n. (62)Since the left eigenvectors f ± k , k = 1 , . . . , n , are linearly independent, the only solution for the n unknowns h η i is the trivial one, i.e., h X j i = h P j i = 0 . It is worth to point out that thisresult simplifies Eqs.(56,57,59).On the other hand, the mean values of the quadratic operators X i , P i , i = 1 , . . . , n , in theextremal state | i can be obtained from evaluating the corresponding quantities for the severalnon-equivalent products of pairs of annihilation B j and creation B † k operators. It is important tomention that these products should have the appropriate order to use the fact that B j annihilates | i and B † k annihilates h | (if the product involves one B j it should be placed to the right while10f it involves one B † k it should be placed to the left). In general, we will get n (2 n + 1) non-equivalent products of pairs of operators B i , B † j , i, j = 1 , . . . , n : n ( n + 1) / products of kind B i B j , j = 1 , . . . , n, i ≤ j ; n ( n + 1) / products of kind B † i B † j , j = 1 , . . . , n, i ≤ j ; n products of kind B † i B j , i, j = 1 , . . . , n . The mean values in the extremal state | i will lead toan inhomogeneous systems of n (2 n + 1) equations, with the same number of unknowns. Whensolving this system we will get h X j i , h P j i , j = 1 , . . . , n , and the mean value of any otherproduct of two canonical operators X i , P j .It is customary nowadays to group the mean values of the quadratic products of the operators X i , P j in the coherent state | z i in a n × n real symmetric matrix σ ( z ) , called covariancematrix, whose elements are given by (remember that η = ( X , . . . , X n , P , . . . , P n ) T ): σ ij ( z ) = h η i η j + η j η i i z − h η i i z h η j i z , i, j = 1 , . . . n. (63)A straightforward calculation leads to σ ij ( z ) = σ ij ( ) = h η i η j + η j η i i ≡ σ ij , (64)where we have used that h η i i = 0 , i = 1 , . . . , n . The conclusion is that the covariance matrixin our coherent state | z i is once again independent from z and depends just of the extremal state | i . Notice that the number of independent matrix elements σ ij ( n (2 n + 1) ) coincides with thenumber of unknowns which are determined from the set of n (2 n + 1) independent equationsassociated to the mean values of the quadratic products of B † j , B k in the extremal state | i .Once the covariance matrix is determined, the generalized uncertainty relation can be eval-uated [33–35] σ i i σ n + i n + i − σ i n + i ≥ , i = 1 , . . . , n, (65)which coincides with the Robertson-Schr¨odinger uncertainty relation (see e.g. [33]).Let us end up this section by calculating the mean value of the Hamiltonian in a givencoherent state | z i . Equation (28) leads to h H i z = h z | H | z i = n X k =1 γ k ω k | z k | + g ′ . (66)In order to get h H i z , let us notice that H = n X j,k =1 γ j γ k ω j ω k B † j B † k B j B k + n X k =1 ω k B † k B k + 2 g ′ n X k =1 γ k ω k B † k B k + g ′ . (67)Thus we get h H i z = n X j,k =1 γ j γ k ω j ω k | z j | | z k | + n X k =1 ω k | z k | + 2 g ′ n X k =1 γ k ω k | z k | + g ′ . (68)Hence, (∆ H ) z = n X k =1 ω k | z k | . (69)Notice that, for one-dimensional systems ( n = 1 ), this expression reduces to the standardone for the harmonic oscillator (see e.g. [1]). 11 .3 Time evolution of the CS Suppose that at t = 0 our system is in a coherent state | z i . Thus, at a later time t > the evolvedstate is found by acting on | z i with the evolution operator of the system U ( t ) = exp( − iHt ) .By making use of Eq.(29) it turns out that U ( t ) = e − ig ′ t n Y k =1 e − iγ k ω k N k . Hence, U ( t ) | z i = e − ig ′ t | z ( t ) , . . . , z n ( t ) i = e − ig ′ t | z ( t ) i , (70)where z j ( t ) = e − iγ j ω j t z j = | z j | e i ( θ j − γ j ω j t ) . Equation (70) implies that a coherent state | z i evolves in time into a new coherent state | z ( t ) i = | z ( t ) , . . . , z n ( t ) i , where the j -th degree offreedom z j ( t ) just rotates at its characteristic frequency ω j (clockwise if γ j = 1 and counter-clockwise if γ j = − ). At this point, it would be interesting to check if our CS belong to the class introduced recentlyby Gazeau and Klauder [36]. Using their notation, for a system with a Hamiltonian H such thatthe ground state energy is zero, the Gazeau-Klauder CS {| J, θ i , J ≥ , −∞ < θ < ∞} obeythe following properties:(a) Continuity: ( J ′ , θ ′ ) → ( J, θ ) ⇒ | J ′ , θ ′ i → | J, θ i .(b) Resolution of unity: = R | J, θ ih J, θ | dµ ( J, θ ) .(c) Temporal stability: e − i H t | J, θ i = | J, θ + ωt i , ω = constant .(d) Action identity: h J, θ |H|
J, θ i = ωJ .Concerning the first property, it is straightforward to check that our coherent states given inEq.(43) are such that | z ′ i → | z i as z ′ → z , i.e., they are continuous in z . As for the second andthird properties, both were explicitly proven in sections 4.1 and 4.3 respectively. It remains justto analyze if it is valid the action identity given in (d). Let us notice first of all that it is valid foreach partial Hamiltonian H k = γ k ω k N k of our system, h z | H k | z i = γ k ω k | z k | , which is time-independent. Therefore, property (d) becomes valid for each degree of freedomseparately and thus it is valid for our global system with the natural identification J k = | z k | , θ k = arg( z k ) so that h z | ( H − g ′ ) | z i = n X k =1 γ k ω k J k . We conclude that our CS of Eq.(43) become as well an n -dimensional generalization of theGazeau-Klauder CS if we express each complex component z k of z in its polar form (the polarcoordinates essentially coincide with the canonical action-angle variables for the correspondingclassical system). 12 Asymmetric Penning trap coherent states
Let us apply now the previous technique to the asymmetric Penning trap. Such an arrangementcan be used to control some quantum mechanical phenomena [37] as well as to perform high-precision measurements of fundamental properties of particles. Moreover, it is a quite naturalsystem to analyze the decoherence taking place due to the unavoidable interaction of the systemwith its environment [38, 39]. Since the asymmetric Penning trap becomes the ideal one whenthe asymmetry parameter vanishes [40–42], it will be straightforward to compare these resultswith those recently obtained for the ideal Penning trap [16] (see also [35, 43, 44]).The Hamiltonian of a charged particle with mass m and charge q in an asymmetric Penningtrap reads H = ~P m + ω c XP y − Y P x ) + m ω x X + ω y Y + ω z Z ) , (71) ω c = qB/m and ω z being the cyclotron and axial frequencies respectively, and the frequencies ω x , ω y are given by ω x = ω c − ω z ε ) , ω y = ω c − ω z − ε ) , (72)where | ε | < is the real asymmetry parameter and we are denoting ~P = ( P x , P y , P z ) T , ~X =( X, Y, Z ) T . Without loosing generality [16], from now on we will assume that m = 1 .As it was seen at section 2, the main role in our treatment is played by the matrix Λ suchthat [ iH, η ] = Λ η . We choose here η = ( X, Y, P x , P y , Z, P z ) T so that Λ = − ω c / ω c / − ω x − ω c / − ω y ω c / − ω z . (73)The eigenvalues ( λ ) of Λ are λ = iω c √ − δ + R = iω , λ = iω c √ − δ − R = iω ,λ = iω z = iω , R = p − δ ) + δ ε , < δ = 2 ω z ω c < , (74)and their corresponding complex conjugate. The right ( u ) and left ( f ) eigenvectors of Λ be-come u +1 = s (cid:18) ω c ( δε + R ) , − iω δε + Rδε + R , iω c ω − δ ) − δε + Rδε + R , , , (cid:19) T ,u +2 = s (cid:18) ω c ( δε − R ) , − iω δε − Rδε − R , iω c ω − δ ) − δε − Rδε − R , , , (cid:19) T ,u +3 = s (cid:18) , , , , − iω z , (cid:19) T , +1 = t (cid:18) ω c R − δε ) , iω c ω [2(1 − δ ) + δε + R ] , − iω c ω (2 − δε + R ) , , , (cid:19) ,f +2 = t (cid:18) ω c − R − δε ) , iω c ω [2(1 − δ ) + δε − R ] , − iω c ω (2 − δε − R ) , , , (cid:19) ,f +3 = t (0 , , , , iω z , , (75)where s j , t j ∈ C , j = 1 , , . The requirement that the right and left eigenvectors be dual toeach other implies s = 14 t (cid:18) δεR (cid:19) , s = 14 t (cid:18) − δεR (cid:19) , s = 12 t . (76)On the other hand, up to some phase factors, the condition imposed by Eqs.(10,11) leads to, t = 1 p i ( f a f c − f b ) , t = 1 p i ( f b − f a f c ) , t = 1 √ ω , (77)where we are denoting f +1 = t ( f a , f b , f c , , , , f +2 = t ( f a , f b , f c , , , in order tosimplify the notation (compare Eq.(75)). Moreover, the crucial signs for us to conclude that ourasymmetric Penning trap Hamiltonian is not positively defined become: γ = 1 , γ = − , γ = 1 . (78)Thus, our annihilation operators take the form (see Eq.(22)): B = L − = t ( f a X − f b Y − f c P x + P y ) ,B = L +2 = t ( f a X + f b Y + f c P x + P y ) , (79) B = L − = t ( − iω Z + P z ) . From these operators and their hermitian conjugates, it is straightforward to identify the α j and β j , j = 1 , , which allow us to find the matrix a such that a α j = β j . Its matrix elements a ij become now: a = − i f a − f a f c + f c , a = a = i f b + f b f c + f c , a = i f c f b − f c f b f c + f c ,a = ω , a = a = a = a = 0 . (80)It can be shown that a , a , a ∈ R + while a is purely imaginary. Thus the extremal statewave function of Eq.(31) acquires the form: φ ( ~x ) = c exp (cid:18) − a x − a y − a xy (cid:19) exp (cid:18) − a z (cid:19) . (81)The associated eigenvalue becomes E , , = ( ω − ω + ω ) / .Concerning the coherent states | z , z , z i , the general treatment developed in section 3 isstraightforwardly applicable, and their explicit expressions are given by Eq.(43) with n = 3 .Their corresponding wave functions are given by φ z ( ~x ) = h ~x | z i = e − i~ Γ · ~ Σ / e i~ Σ · ~x φ ( x − Γ , y − Γ , z − Γ ) , (82)14here ~ Γ = 2 it f c Re [ z ] − it f c Re [ z ] − t Im [ z ] − t Im [ z ] − t Im [ z ] , ~ Σ = 2 t f a Im [ z ] + t f a Im [ z ] − it f b Re [ z ] + it f b Re [ z ] t ω Re [ z ] . (83)The mean values h X j i z , h P j i z , immediately follow from Eqs.(56, 59) with h X j i = h P j i =0 , i.e., h X j i z = Γ j , h P j i z = Σ j , j = 1 , , . (84)As for the mean values of the quadratic operators in the extremal state, we have solved thesystem of equations arising from the null mean values of the products of pairs of annihilation B j and creation B † k operators. We get h X i = 12 a , h P x i = 12 (cid:18) a − a a (cid:19) , h Y i = 12 a , h P y i = 12 (cid:18) a − a a (cid:19) , (85) h Z i = 12 a , h P z i = 12 a , and the crossed products h XP x i = i , h XP y i = i a a , h XP z i = 0 , h Y P x i = i a a , h Y P y i = i , h Y P z i = 0 , (86) h ZP x i = 0 , h ZP y i = 0 , h ZP z i = i . Therefore, using Eqs.(58, 60) we get the Heisenberg uncertainty relationships (∆ X ) z (∆ P x ) z = (∆ Y ) z (∆ P y ) z = 14 (cid:18) | a | a a (cid:19) ≥ , (87) (∆ Z ) z (∆ P z ) z = 14 , while Eq.(69) with n = 3 gives the mean square deviation for the Hamiltonian.Once we have calculated the mean values of the quadratic products given in Eqs.(85,86),it is straightforward to evaluate the covariance matrix elements of Eq.(64). With the ordering η = ( X, Y, P x , P y , Z, P z ) T , it is obtained: σ = (∆ X ) ia a Y ) ia a ia a (∆ P x ) ia a P y ) Z )
00 0 0 0 0 (∆ P z ) . (88)15otice that this covariance matrix is non-diagonal. However, since σ = σ = σ = 0 , itturns out that the generalized uncertainty relations of Eq.(65) reduce to the Heisenberg uncer-tainty relations given in Eq.(87).A plot of (∆ X ) z (∆ P x ) z as a function of the parameters ε and δ is given in Fig.1. As it can beseen from Eqs.(80,87) and from Fig.1, the coherent states minimize the Heisenberg uncertaintyrelationship for ε = 0 , which coincides with the results recently obtained for the ideal Penningtrap [16]. However, for ε = 0 it turns out that (∆ X ) z (∆ P x ) z > / . Notice that the same plotwill appear for the uncertainty product (∆ Y ) z (∆ P y ) z .Figure 1: Heisenberg uncertainty relationship (∆ X ) z (∆ P x ) z for the asymmetric Penning trap coherent states asfunction of the real dimensionless parameters | ε | < , < δ < . In this work we have proposed a systematic technique to find the CS for systems governed byquadratic Hamiltonians in the trap regime. To do this, we introduced a prescription to identifyin a simple way the appropriate ladder operators which play the same role as the annihilationand creation operators for the -dimensional harmonic oscillator. These operators allowed usto generate the eigenvectors and eigenvalues for the Hamiltonian departing from the extremalstate, the analogue of the ground state although it is not necessarily an eigenstate associated tothe lowest possible eigenvalue. The explicit expression for the extremal state wave function wasas well explicitly calculated.For systems governed by this kind of Hamiltonians the two algebraic CS definitions (ei-ther as simultaneous eigenstates of the annihilation operators or as resulting from the action ofthe displacement operator onto the extremal state) lead to the same set of states. The explicitexpression for the corresponding wave functions has been also derived.We have calculated explicitly the mean values of the position and momentum operatorsin an arbitrary coherent state. Moreover, we have provided as well a prescription to obtain16lgebraically, by solving a linear systems of equations, the mean values of the quadratic productsof these operators in the CS.Through this method we have found the asymmetric Penning trap coherent states and wehave explored some of their physical properties. In particular, it is worth to point out that, ingeneral, they do not minimize the Heisenberg uncertainty relationship. The differences fromthe minimum are induced by the deviations of the axial symmetry which the ideal Penning traphas (measured by the asymmetry parameter ε ). Acknowledgments
The authors acknowledge the support of Conacyt.
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